Quasi-asymptotically conical Calabi-Yau manifolds

TL;DR

This paper constructs new examples of quasi-asymptotically conical Calabi-Yau manifolds, expanding the known classes by developing a compactification framework and solving complex Monge-Ampère equations.

Contribution

It introduces a natural compactification of QAC-spaces and defines QAC-metrics via a Lie algebra, enabling the construction of new Calabi-Yau metrics.

Findings

Constructed new QAC Calabi-Yau examples

Developed a compactification with fibred corners

Solved complex Monge-Ampère equations for Ricci-flat metrics

Abstract

We construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). We do so by first providing a natural compactification of QAC-spaces by manifolds with fibred corners and by giving a definition of QAC-metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on QAC-spaces developed by the second author and Mazzeo, we can in many instances obtain K\"ahler QAC-metrics having Ricci potential decaying sufficiently fast at infinity. This allows us to obtain QAC Calabi-Yau metrics in the K\"ahler classes of these metrics by solving a corresponding complex Monge-Amp\`ere equation.